Conditional Probability

Definition

The conditional probability of event $A$ given that event $B$ has occurred:

$P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad P(B) > 0$

Read as: "the probability of A given B"

Using Two-Way Tables

Two-way tables are the easiest way to compute conditional probabilities.

Example: Students by grade and sport participation:

| | Plays Sport | No Sport | Total | |---|---|---|---| | Grade 9 | 40 | 60 | 100 | | Grade 10 | 55 | 45 | 100 | | Total | 95 | 105 | 200 |

$P(\text{Sport} | \text{Grade 9}) = \frac{40}{100} = 0.40$ $P(\text{Grade 9} | \text{Sport}) = \frac{40}{95} \approx 0.421$

Notice: $P(A|B) \neq P(B|A)$ in general!

Tree Diagrams

Tree diagrams visualize sequential events:

1. First branch: probabilities of first event
2. Second branch: conditional probabilities given the first event
3. Multiply along branches for joint probabilities
4. Add final probabilities for total

General Multiplication Rule

$P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$

Bayes' Theorem

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Expanded form: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B|A) \cdot P(A) + P(B|A^c) \cdot P(A^c)}$

Law of Total Probability

$P(B) = P(B|A) \cdot P(A) + P(B|A^c) \cdot P(A^c)$

AP Tip: On the AP exam, always show your work with conditional probability. Label what each probability represents and use proper notation $P(A|B)$.